Infinitely many solutions for elliptic system with Hamiltonian type

In this paper, we use Legendre-Fenchel transform and a space decomposition to carry out Fountain theorem and dual Fountain theorem for the following elliptic system of Hamiltonian type: $$\{\begin{array}{l}\begin{array}{rlrl}{\displaystyle -\mathrm{\Delta }u}& {\displaystyle =H_v(u,v)}& {\displaystyle }& {\displaystyle \text{in }\mathrm{\Omega },}\\ {\displaystyle -\mathrm{\Delta }v}& {\displaystyle =H_u(u,v)}& {\displaystyle }& {\displaystyle \text{in }\mathrm{\Omega },}\\ {\displaystyle u,v}& {\displaystyle =0}& {\displaystyle }& {\displaystyle \text{on }\mathrm{\partial }\mathrm{\Omega },}\end{array}\end{array}$$ where $N\ge 1$, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain and $H\in C^1({\mathbb{R}}^2)$ is strictly convex, even and subcritical. We mainly present two results: (i) When $H$ is superlinear, the system has infinitely many solutions, whose energies tend to infinity. (ii) When $H$ is sublinear, the system has infinitely many solutions, whose energies are negative and tend to 0. As a byproduct, the Lane-Emden system under subcritical growth has infinitely many solutions.